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Journal articles on the topic 'Stochastic'

To see the other types of publications on this topic, follow the link: Stochastic.

Author: Grafiati

Published: 4 June 2021

Last updated: 8 June 2024

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1

Engelbert, H. J., and V. P. Kurenok. "On Multidimensional SDEs Without Drift and with A Time-Dependent Diffusion Matrix." Georgian Mathematical Journal 7, no. 4 (December 2000): 643–64. http://dx.doi.org/10.1515/gmj.2000.643.

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Abstract We study multidimensional stochastic equations where x o is an arbitrary initial state, W is a d-dimensional Wiener process and is a measurable diffusion coefficient. We give sufficient conditions for the existence of weak solutions. Our main result generalizes some results obtained by A. Rozkosz and L. Słomiński [Stochastics Stochasties Rep. 42: 199–208, 1993] and T. Senf [Stochastics Stochastics Rep. 43: 199–220, 1993] for the existence of weak solutions of one-dimensional stochastic equations and also some results by A. Rozkosz and L. Słomiński [Stochastic Process. Appl. 37: 187–197, 1991], [Stochastic Process. Appl. 68: 285–302, 1997] for multidimensional equations. Finally, we also discuss the homogeneous case.

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2

Luo, Mei-Ju, and Yuan Lu. "Properties of Expected Residual Minimization Model for a Class of Stochastic Complementarity Problems." Journal of Applied Mathematics 2013 (2013): 1–7. http://dx.doi.org/10.1155/2013/497586.

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Expected residual minimization (ERM) model which minimizes an expected residual function defined by an NCP function has been studied in the literature for solving stochastic complementarity problems. In this paper, we first give the definitions of stochasticP-function, stochasticP0-function, and stochastic uniformlyP-function. Furthermore, the conditions such that the function is a stochasticPP0-function are considered. We then study the boundedness of solution set and global error bounds of the expected residual functions defined by the “Fischer-Burmeister” (FB) function and “min” function. The conclusion indicates that solutions of the ERM model are robust in the sense that they may have a minimum sensitivity with respect to random parameter variations in stochastic complementarity problems. On the other hand, we employ quasi-Monte Carlo methods and derivative-free methods to solve ERM model.

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3

IMKELLER, PETER, and ADAM HUGH MONAHAN. "CONCEPTUAL STOCHASTIC CLIMATE MODELS." Stochastics and Dynamics 02, no. 03 (September 2002): 311–26. http://dx.doi.org/10.1142/s0219493702000443.

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From July 9 to 11, 2001, 50 researchers from the fields of climate dynamics and stochastic analysis met in Chorin, Germany, to discuss the idea of stochastic models of climate. The present issue of Stochastics and Dynamics collects several papers from this meeting. In this introduction to the volume, the idea of simple conceptual stochastic climate models is introduced amd recent results in the mathematically rigorous development and analysis of such models are reviewed. As well, a brief overview of the application of ideas from stochastic dynamics to simple models of the climate system is given.

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4

LIUM, ARNT-GUNNAR, TEODOR GABRIEL CRAINIC, and STEIN W. WALLACE. "CORRELATIONS IN STOCHASTIC PROGRAMMING: A CASE FROM STOCHASTIC SERVICE NETWORK DESIGN." Asia-Pacific Journal of Operational Research 24, no. 02 (April 2007): 161–79. http://dx.doi.org/10.1142/s0217595907001206.

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Deterministic models, even if used repeatedly, will not capture the essence of planning in an uncertain world. Flexibility and robustness can only be properly valued in models that use stochastics explicitly, such as stochastic optimization models. However, it may also be very important to capture how the random phenomena are related to one another. In this article we show how the solution to a stochastic service network design model depends heavily on the correlation structure among the random demands. The major goal of this paper is to discuss why this happens, and to provide insights into the effects of correlations on solution structures. We illustrate by an example.

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5

Liu, Caixia, Yingqi Zhang, and Huixia Sun. "Finite-TimeH∞Filtering for Singular Stochastic Systems." Journal of Applied Mathematics 2012 (2012): 1–16. http://dx.doi.org/10.1155/2012/615790.

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This paper addresses the problem of finite-timeH∞filtering for one family of singular stochastic systems with parametric uncertainties and time-varying norm-bounded disturbance. Initially, the definitions of singular stochastic finite-time boundedness and singular stochasticH∞finite-time boundedness are presented. Then, theH∞filtering is designed for the class of singular stochastic systems with or without uncertain parameters to ensure singular stochastic finite-time boundedness of the filtering error system and satisfy a prescribedH∞performance level in some given finite-time interval. Furthermore, sufficient criteria are presented for the solvability of the filtering problems by employing the linear matrix inequality technique. Finally, numerical examples are given to illustrate the validity of the proposed methodology.

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6

Zhang, Yingqi, Wei Cheng, Xiaowu Mu, and Caixia Liu. "Stochasticℋ∞Finite-Time Control of Discrete-Time Systems with Packet Loss." Mathematical Problems in Engineering 2012 (2012): 1–15. http://dx.doi.org/10.1155/2012/897481.

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This paper investigates the stochastic finite-time stabilization andℋ∞control problem for one family of linear discrete-time systems over networks with packet loss, parametric uncertainties, and time-varying norm-bounded disturbance. Firstly, the dynamic model description studied is given, which, if the packet dropout is assumed to be a discrete-time homogenous Markov process, the class of discrete-time linear systems with packet loss can be regarded as Markovian jump systems. Based on Lyapunov function approach, sufficient conditions are established for the resulting closed-loop discrete-time system with Markovian jumps to be stochasticℋ∞finite-time boundedness and then state feedback controllers are designed to guarantee stochasticℋ∞finite-time stabilization of the class of stochastic systems. The stochasticℋ∞finite-time boundedness criteria can be tackled in the form of linear matrix inequalities with a fixed parameter. As an auxiliary result, we also give sufficient conditions on the robust stochastic stabilization of the class of linear systems with packet loss. Finally, simulation examples are presented to illustrate the validity of the developed scheme.

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7

Polyakova, A. Yu. "Feat ures of continuing formati on of stochastic cult ure of schoolchildren in the conditions of dista nce learning." Informatics in school, no. 6 (September 25, 2021): 39–48. http://dx.doi.org/10.32517/2221-1993-2021-20-6-39-48.

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The article describes the essence of continuity in the formation of a stochastic culture of students in the conditions of distance learning. The author’s definition of the concept “continuity in the formation of stochastic culture of students of a general education school in the conditions of distance learning” is given. There are the basic conditions of a teacher’s possession of the highest level of stochastic culture, due to which the formation of an integral personal quality, which is a generalized indicator of stochastic competence, takes place in schoolchildren. Modern information and communication technologies used in a series of developed distance classes in statistics, combinatorics and probability theory for grade 11 students are proposed. These ICT contribute to the successive formation of elements of stochastic culture in the conditions of distance learning. The article emphasizes that the use of ICT in teaching stochastics is an effective methodological tool, through which significant results can be achieved in the educational process.

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8

Hu, Peng, and Chengming Huang. "The StochasticΘ-Method for Nonlinear Stochastic Volterra Integro-Differential Equations." Abstract and Applied Analysis 2014 (2014): 1–13. http://dx.doi.org/10.1155/2014/583930.

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The stochasticΘ-method is extended to solve nonlinear stochastic Volterra integro-differential equations. The mean-square convergence and asymptotic stability of the method are studied. First, we prove that the stochasticΘ-method is convergent of order1/2in mean-square sense for such equations. Then, a sufficient condition for mean-square exponential stability of the true solution is given. Under this condition, it is shown that the stochasticΘ-method is mean-square asymptotically stable for every stepsize if1/2≤θ≤1and when0≤θ<1/2, the stochasticΘ-method is mean-square asymptotically stable for some small stepsizes. Finally, we validate our conclusions by numerical experiments.

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9

Sankar, T. S., S. A. Ramu, and R. Ganesan. "Stochastic Finite Element Analysis for High Speed Rotors." Journal of Vibration and Acoustics 115, no. 1 (January 1, 1993): 59–64. http://dx.doi.org/10.1115/1.2930315.

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The general problem of the dynamic response of highspeed rotors is considered in which certain system parameters may have a spatial stochastic variation. In particular the elastic modulus and mass density of a rotating shaft are described through one dimensional stochastic field functions so that the imperfections in manufacture and measurement can be accounted for. The stochastic finite element method is developed so that the variability of the response of the rotor can be interpreted in terms of the variation of the material property. As an illustration the whirl speed analysis is performed to determine the stochastics of whirl speeds and modes through the solution of a random eigenvalue problem associated with a non self-adjoint system. Numerical results are also presented.

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10

TRIPŞA, Florența-Violeta, and Ana-Maria LUCA (RȊTEA). "STOCHASTIC APPROXIMATION FOR RELIABILITY PROBLEMS." SCIENTIFIC RESEARCH AND EDUCATION IN THE AIR FORCE 18, no. 1 (June 24, 2016): 497–500. http://dx.doi.org/10.19062/2247-3173.2016.18.1.68.

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11

Gozzi, Ennio. "Stochastic and Non-Stochastic Supersymmetry." Progress of Theoretical Physics Supplement 111 (1993): 115–50. http://dx.doi.org/10.1143/ptps.111.115.

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12

Veraart, Almut E. D., and Luitgard A. M. Veraart. "Stochastic volatility and stochastic leverage." Annals of Finance 8, no. 2-3 (May 21, 2010): 205–33. http://dx.doi.org/10.1007/s10436-010-0157-3.

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13

Lykova, K. G. "METHODOLOGY OF FORMING A STOCHASTIC WORLDVIEW WHILE STUDYING THE SECTION "RANDOM EVENTS. VEROBILITIES"." Educational Psychology in Polycultural Space 56, no. 4 (2021): 67–77. http://dx.doi.org/10.24888/2073-8439-2021-56-4-67-77.

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The article contains general provisions, characterizing the methodology of forming stochastic worldview of high school students. The methodology is carried out in accordance with the developed worldview significant educational material, accompanied by the support of specially selected digital technology, and in accordance with the main stages of the formation of worldview while teaching mathematics. The main stages of the implementation of the holistic process of learning stochasticity are: preparatory, problem-research, implementation and correctional. It is supposed that the methodology of forming stochastic outlook will be most effective if during the learning process students of 10–11 grades have sustainable learning and cognitive motives and motives for self-education, the system of solid subject knowledge of stochastics, the mastery of probabilistic-statistical methods; the change of activities is observed: reproductive to learning and research activity, productive to problem-creative activity. The article argues that the methodology of forming stochastic worldview of high school students, including the support and accompaniment of digital educational technologies, has a positive impact on motivation and cognitive interest to learning, changing attitudes towards stochastics, expanding scientific horizons. Through consistent work with the established stages of the formation of stochastic worldview in teaching mathematics, the realization of the integrity of motivational, value and cognitive constructs is achieved. In this article this process will be considered by studying the section “Random events. Probabilities”. At the first (preparatory) stage the basic concepts of the section are introduced, the skills of their understanding and application in learning activities are developed. At the second (problem-searching) stage the theorems and formulas of the section are examined, tools for problem solving are formed, the understanding of methods of action with stochastic situations is improved. At the third (realization) stage purposeful cognitive-research and problem-creative activity of senior pupils is organized, including development and consolidation of knowledge, abilities and skills of work with stochastic situations. At the fourth (correctional) stage reflexive activity of students is organized, the teacher determines the degree of expression of outlook qualities and reference points of high school students.

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14

Sihotang, Hengki Tamando, Syahril Efendi, Muhammad Zarlis, and Herman Mawengkang. "Data driven approach for stochastic data envelopment analysis." Bulletin of Electrical Engineering and Informatics 11, no. 3 (June 1, 2022): 1497–504. http://dx.doi.org/10.11591/eei.v11i3.3660.

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Decision making based on data driven deals with a large amount of data will evaluate the process's effectiveness. Evaluate effectiveness in this paper is measure of performance efficiency of data envelopment analysis (DEA) method in this study is the approach with uncertainty problems. This study proposed a new method called the robust stochastic DEA (RSDEA) to approach performance efficiency in tackling uncertainty problems (i.e., stochastic and robust optimization). The RSDEA method develops to combine the stochastics DEA (SDEA) formulation method and Robust Optimization. The numerical example demonstrates the performance efficiency of the proposed formulation method, with the results performing confirmed that the efficiency value is 89%.

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15

MTW and H. Kunita. "Stochastic Flows and Stochastic Differential Equations." Journal of the American Statistical Association 93, no. 443 (September 1998): 1251. http://dx.doi.org/10.2307/2669903.

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16

Krylov, Nicolai. "Stochastic flows and stochastic differential equations." Stochastics and Stochastic Reports 51, no. 1-2 (November 1994): 155–58. http://dx.doi.org/10.1080/17442509408833949.

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17

Bayraktar, Erhan, and Jiaqi Li. "Stochastic Perron for stochastic target games." Annals of Applied Probability 26, no. 2 (April 2016): 1082–110. http://dx.doi.org/10.1214/15-aap1112.

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18

Jacka, S. D., and H. Kunita. "Stochastic Flows and Stochastic Differential Equations." Journal of the Royal Statistical Society. Series A (Statistics in Society) 155, no. 1 (1992): 175. http://dx.doi.org/10.2307/2982680.

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19

De Falco, Diego. "Lévy's stochastic area in stochastic mechanics." Letters in Mathematical Physics 14, no. 1 (July 1987): 41–45. http://dx.doi.org/10.1007/bf00403468.

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20

Birge, John R., and M. A. H. Dempstert. "Stochastic programming approaches to stochastic scheduling." Journal of Global Optimization 9, no. 3-4 (December 1996): 417–51. http://dx.doi.org/10.1007/bf00121682.

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21

Bayraktar, Erhan, and Jiaqi Li. "Stochastic Perron for Stochastic Target Problems." Journal of Optimization Theory and Applications 170, no. 3 (May 31, 2016): 1026–54. http://dx.doi.org/10.1007/s10957-016-0958-2.

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22

Hu, Wei. "Stochastic Finite-Time Stability for Stochastic Nonlinear Systems with Stochastic Impulses." Symmetry 14, no. 4 (April 14, 2022): 817. http://dx.doi.org/10.3390/sym14040817.

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In this paper, some novel stochastic finite-time stability criteria for stochastic nonlinear systems with stochastic impulse effects are established. The results in this paper blackgeneralized the related results in from two aspects: 1. the model in is the deterministic systems, which means that the noise effect that can be described as a symmetric Markov process Brownian motion is considered in our models; 2. the stochastic finite-time stability criterion is established in this paper, not the asymptotic stability and the input-to-state stability that are studied in the form literature. Finally, an example is given to show the significance blackand usefulness of our results.

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23

Zhao, Wenqiang, and Yangrong Li. "Existence of Random Attractors for ap-Laplacian-Type Equation with Additive Noise." Abstract and Applied Analysis 2011 (2011): 1–21. http://dx.doi.org/10.1155/2011/616451.

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We first establish the existence and uniqueness of a solution for a stochasticp-Laplacian-type equation with additive white noise and show that the unique solution generates a stochastic dynamical system. By using the Dirichlet forms of Laplacian and an approximation procedure, the nonlinear obstacle, arising from the additive noise is overcome when we make energy estimate. Then, we obtain a random attractor for this stochastic dynamical system. Finally, under a restrictive assumption on the monotonicity coefficient, we find that the random attractor consists of a single point, and therefore the system possesses a unique stationary solution.

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24

Petuhov, Sergey. "MUSIC, UNIVERSALS OF GENOMIC DNAs, GENETIC INTELLIGENCE, AND THE CONCEPT OF A STOCHASTIC SYSTEM OF PROTOREGULATION." Medicine and Art 1, no. 2 (August 10, 2023): 40–57. http://dx.doi.org/10.60042/2949-2165-2023-1-2-40-57.

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This work is devoted to the structural links between the musical harmony of the Pythagorean fifth scales and the system of hydrogen bonds in genetic molecules DNAs. Hidden regularities in the stochastic organization of hydrogen bond sequences in the genomic DNAs of higher and lower organisms have been discovered. The regularities described are candidates for the role of universal regularities of biological evolution. An idea is advanced on the fundamental role of stochastic regularities in the genetically inherited organization of biological bodies. In the presented study, the author's method of binary stochastics hierarchy for genetic and physiological studies was used.

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25

Vrkoč, Ivo. "Weak averaging of stochastic evolution equations." Mathematica Bohemica 120, no. 1 (1995): 91–111. http://dx.doi.org/10.21136/mb.1995.125891.

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26

Březková, L., M. Starý, and P. Doležal. "The real-time stochastic flow forecast." Soil and Water Research 5, No. 2 (May 24, 2010): 49–57. http://dx.doi.org/10.17221/13/2009-swr.

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In the Czech Republic, deterministic flow forecasts with the lead time of 48 hours, calculated by rainfall-runoff models for basins of a size of several hundreds to thousands square kilometers, are nowadays a common part of the operational hydrological service. The Czech Hydrometeorological Institute (CHMI) issues daily the discharge forecast for more than one hundred river profiles. However, the causal rainfall is a random process more than a deterministic one, therefore the deterministic discharge forecast based on one precipitation prediction is a significant simplification of the reality. Since important decisions must be done during the floods, it is necessary to take into account the indeterminity of the input meteorological data and to express the uncertainty of the resulting discharge forecast. In the paper, a solution of this problem is proposed. The time series of the input precipitation prediction data have been generated repeatedly (by the Monte Carlo method) and, subsequently, the set of discharge forecasts based on the repeated hydrological model simulations has been obtained and statistically evaluated. The resulting output can be, for example, the range of predicted peak discharges, the peak discharge exceeding curve or the outflow volume exceeding curve. The properties of the proposed generator have been tested with acceptable results on several flood events which occurred over the last years in the upper part of the Dyje catchment (Podhradí closing profile). The rainfall-runoff model HYDROG, which has been in operation in CHMI since 2003, was used for hydrological simulation.

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27

Sun, Huiying, Meng Li, Shenglin Ji, and Long Yan. "Stability and Linear Quadratic Differential Games of Discrete-Time Markovian Jump Linear Systems with State-Dependent Noise." Mathematical Problems in Engineering 2014 (2014): 1–11. http://dx.doi.org/10.1155/2014/265621.

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We mainly consider the stability of discrete-time Markovian jump linear systems with state-dependent noise as well as its linear quadratic (LQ) differential games. A necessary and sufficient condition involved with the connection between stochasticTn-stability of Markovian jump linear systems with state-dependent noise and Lyapunov equation is proposed. And using the theory of stochasticTn-stability, we give the optimal strategies and the optimal cost values for infinite horizon LQ stochastic differential games. It is demonstrated that the solutions of infinite horizon LQ stochastic differential games are concerned with four coupled generalized algebraic Riccati equations (GAREs). Finally, an iterative algorithm is presented to solve the four coupled GAREs and a simulation example is given to illustrate the effectiveness of it.

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28

Field, R. V., M. Grigoriu, and J. M. Emery. "On the efficacy of stochastic collocation, stochastic Galerkin, and stochastic reduced order models for solving stochastic problems." Probabilistic Engineering Mechanics 41 (July 2015): 60–72. http://dx.doi.org/10.1016/j.probengmech.2015.05.002.

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29

Leemans, Sander J. J., Wil M. P. van der Aalst, Tobias Brockhoff, and Artem Polyvyanyy. "Stochastic process mining: Earth movers’ stochastic conformance." Information Systems 102 (December 2021): 101724. http://dx.doi.org/10.1016/j.is.2021.101724.

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30

Arvanitis, Stelios. "Stochastic dominance efficient sets and stochastic spanning." Decisions in Economics and Finance 44, no. 1 (March 11, 2021): 401–9. http://dx.doi.org/10.1007/s10203-021-00325-y.

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31

Lu, Xiang, Gunter Meissner, and Hong Sherwin. "A Unified Stochastic Volatility—Stochastic Correlation Model." Journal of Mathematical Finance 10, no. 04 (2020): 679–96. http://dx.doi.org/10.4236/jmf.2020.104039.

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32

Ohzeki, Masayuki. "Stochastic gradient method with accelerated stochastic dynamics." Journal of Physics: Conference Series 699 (March 2016): 012019. http://dx.doi.org/10.1088/1742-6596/699/1/012019.

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33

Klimontovich, Yu L. "What are stochastic filtering and stochastic resonance?" Uspekhi Fizicheskih Nauk 169, no. 1 (1999): 39. http://dx.doi.org/10.3367/ufnr.0169.199901d.0039.

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34

Ai, Bao-quan, and Liang-gang Liu. "Stochastic resonance in a stochastic bistable system." Journal of Statistical Mechanics: Theory and Experiment 2007, no. 02 (February 23, 2007): P02019. http://dx.doi.org/10.1088/1742-5468/2007/02/p02019.

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35

Klimontovich, Yu L. "What are stochastic filtering and stochastic resonance?" Physics-Uspekhi 42, no. 1 (January 31, 1999): 37–44. http://dx.doi.org/10.1070/pu1999v042n01abeh000445.

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36

Rice, Sean H., and Anthony Papadopoulos. "Evolution with Stochastic Fitness and Stochastic Migration." PLoS ONE 4, no. 10 (October 9, 2009): e7130. http://dx.doi.org/10.1371/journal.pone.0007130.

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37

Kisielewicz, Michal. "Set-valued stochastic intergrals and stochastic inclutions1." Stochastic Analysis and Applications 15, no. 5 (January 1997): 783–800. http://dx.doi.org/10.1080/07362999708809507.

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38

Nemirovski, A., A. Juditsky, G. Lan, and A. Shapiro. "Robust Stochastic Approximation Approach to Stochastic Programming." SIAM Journal on Optimization 19, no. 4 (January 2009): 1574–609. http://dx.doi.org/10.1137/070704277.

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39

Morita, Hiroshi, and Hiroaki Ishii. "A stochastic improvement method for stochastic programming." Computational Statistics & Data Analysis 14, no. 4 (November 1992): 477–87. http://dx.doi.org/10.1016/0167-9473(92)90063-l.

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40

Rhodes, Remi. "Stochastic Homogenization of Reflected Stochastic Differential Equations." Electronic Journal of Probability 15 (2010): 989–1023. http://dx.doi.org/10.1214/ejp.v15-776.

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41

Higle, J. L., B. Rayco, and S. Sen. "Stochastic scenario decomposition for multistage stochastic programs." IMA Journal of Management Mathematics 21, no. 1 (June 25, 2009): 39–66. http://dx.doi.org/10.1093/imaman/dpp001.

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42

Du, Shouqiang, Maolin Che, and Yimin Wei. "Stochastic structured tensors to stochastic complementarity problems." Computational Optimization and Applications 75, no. 3 (October 22, 2019): 649–68. http://dx.doi.org/10.1007/s10589-019-00144-3.

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43

Sun, Ran, and Yueyue Fan. "Stochastic OD demand estimation using stochastic programming." Transportation Research Part B: Methodological 183 (May 2024): 102943. http://dx.doi.org/10.1016/j.trb.2024.102943.

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44

Hu, Peng, and Chengming Huang. "Stability of stochasticθ-methods for stochastic delay integro-differential equations." International Journal of Computer Mathematics 88, no. 7 (May 2011): 1417–29. http://dx.doi.org/10.1080/00207160.2010.509430.

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45

Williams, David. "STOCHASTIC CALCULUS-A PRACTICAL INTRODUCTION (Probability and Stochastics Series 3)." Bulletin of the London Mathematical Society 30, no. 4 (July 1998): 433–34. http://dx.doi.org/10.1112/s0024609397223805.

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46

Lu, Jin-Biao, Zhi-Jiang Liu, Dmitry Tulenty, Liudmila Tsvetkova, and Sebastian Kot. "Implementation of Stochastic Analysis in Corporate Decision-Making Models." Mathematics 9, no. 9 (May 4, 2021): 1041. http://dx.doi.org/10.3390/math9091041.

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The stochastic approach as a method for modeling factor systems of interrelationships of economic activity aspects allows minimizing managerial errors against the background of company growth and expansion of operating activities. The purpose of this study is to form a decision-making model to ensure the financial competitiveness of enterprises in the context of stochastic analysis. This study demonstrates stochastic analysis implementation in companies of the 2nd and 3rd degrees of internationalization based on multiple regression and factorial analysis of variance. The practical basis of the study was Chinese and Russian mining enterprises that enter highly competitive markets and therefore should avoid mistakes in decision-making as much as possible. The model of financial competitiveness proposed in the article demonstrates the best ways to introduce stochastics in companies to optimize their overall productivity, regardless of the country of origin. In a practical sense, research on reducing managerial mistakes allows enterprises to have financial success even in the turbulent conditions of today’s global market, regardless of the company’s jurisdiction.

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47

Ovchinnikov, Igor V., Wenyuan Li, Yuquan Sun, Andrew E. Hudson, Karlheinz Meier, Robert N. Schwartz, and Kang L. Wang. "Criticality or Supersymmetry Breaking?" Symmetry 12, no. 5 (May 12, 2020): 805. http://dx.doi.org/10.3390/sym12050805.

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In many stochastic dynamical systems, ordinary chaotic behavior is preceded by a full-dimensional phase that exhibits 1/f-type power spectra and/or scale-free statistics of (anti)instantons such as neuroavalanches, earthquakes, etc. In contrast with the phenomenological concept of self-organized criticality, the recently found approximation-free supersymmetric theory of stochastics (STS) identifies this phase as the noise-induced chaos (N-phase), i.e., the phase where the topological supersymmetry pertaining to all stochastic dynamical systems is broken spontaneously by the condensation of the noise-induced (anti)instantons. Here, we support this picture in the context of neurodynamics. We study a 1D chain of neuron-like elements and find that the dynamics in the N-phase is indeed featured by positive stochastic Lyapunov exponents and dominated by (anti)instantonic processes of (creation) annihilation of kinks and antikinks, which can be viewed as predecessors of boundaries of neuroavalanches. We also construct the phase diagram of emulated stochastic neurodynamics on Spikey neuromorphic hardware and demonstrate that the width of the N-phase vanishes in the deterministic limit in accordance with STS. As a first result of the application of STS to neurodynamics comes the conclusion that a conscious brain can reside only in the N-phase.

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48

Buckdahn, Rainer, and Jin Ma. "Pathwise stochastic Taylor expansions and stochastic viscosity solutions for fully nonlinear stochastic PDEs." Annals of Probability 30, no. 3 (July 2002): 1131–71. http://dx.doi.org/10.1214/aop/1029867123.

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Yang, Hua, and Feng Jiang. "Stochasticθ-Methods for a Class of Jump-Diffusion Stochastic Pantograph Equations with Random Magnitude." Scientific World Journal 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/589167.

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Abstract:

This paper is concerned with the convergence of stochasticθ-methods for stochastic pantograph equations with Poisson-driven jumps of random magnitude. The strong order of the convergence of the numerical method is given, and the convergence of the numerical method is obtained. Some earlier results are generalized and improved.

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Wei, Jinlong, Bin Liu, Rongrong Tian, and Liang Ding. "Stochastic Entropy Solutions for Stochastic Scalar Balance Laws." Entropy 21, no. 12 (November 22, 2019): 1142. http://dx.doi.org/10.3390/e21121142.

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We are concerned with the initial value problem for a multidimensional balance law with multiplicative stochastic perturbations of Brownian type. Using the stochastic kinetic formulation and the Bhatnagar-Gross-Krook approximation, we prove the uniqueness and existence of stochastic entropy solutions. Furthermore, as applications, we derive the uniqueness and existence of the stochastic entropy solution for stochastic Buckley-Leverett equations and generalized stochastic Burgers type equations.

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